English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

This paper concerns cluster algebras with principal coefficients A(S,M)associated to bordered surfaces (S,M), and is a companion to a concurrent workof the authors with Schiffler [MSW2]. Given any (generalized) arc or loop inthe surface -- with or without self-intersections -- we associate an element of(the fraction field of) A(S,M), using products of elements of PSL_2(R). We givea direct proof that our matrix formulas for arcs and loops agree with thecombinatorial formulas for arcs and loops in terms of matchings, which weregiven in [MSW, MSW2]. Finally, we use our matrix formulas to prove skeinrelations for the cluster algebra elements associated to arcs and loops. Ourmatrix formulas and skein relations generalize prior work of Fock and Goncharov[FG1, FG2, FG3], who worked in the coefficient-free case. The results of thispaper will be used in [MSW2] in order to show that certain collections of arcsand loops comprise a vector-space basis for A(S,M).

Matrix formulae and skein relations for cluster algebras from surfaces